Interesting. I guess the dev chart that shows "murder" at -5 inf is referring to killing non wanted ships. Otherwise it would contradict what you're saying. So...i didn't run any tests on major faction bounties, beside testing that the crime report take those into account correctly. simply for the reason my testing system (low pop, only 2 factions, only one station) has no faction with major faction allegiance. so, feel free to test!
as you can see from my chart, the effect of killing ships influence drop is tiny compared to the effect of a single bounty claim.
in the test system the relation was around 30:1, e.g. for killing 30 ships of a faction, you'd need to do 1 bountie-claim transaction to even the influence loss. i'm sure this number is that high, because influence of the controlling faction is high (normally above 80%), and as others have found out it is hard to influence minor factions with high influence (diminishing return).
but however this is on your system, i think it is safe to say that influence gain from single bounty-transaction is always higher than influence loss/gain from a single ship kill. so shooting wanted ships of all other faction and cashing in only "your" bounties, as often as possible, should raise your influence massively.
really bad news for anarchistic minor factions, though. those don't give out bountie claims.
To help Faction A:
Kill non wanted ships, but not Faction A ships. Where the inf lost by other factions goes is somewhat uncontrollable in systems with 3 or more factions.
Kill wanted ships of other factions, and only cash in bounties paid by Faction A.
To hurt Faction A:
Kill non wanted Faction A ships.
Kill any wanted ships, but do not cash in bounties paid by Faction A.
Finally, for INF, it doesn't matter where in the system you cash in the bounties, only the faction paying the bounty matters.
For REP, you gain rep with the faction that controls the station where you turn in the bounties.